Nonemptiness problems of plane square tiling with two colors

Abstract

This investigation studies nonemptiness problems of plane square tiling. In the edge coloring (or Wang tiles) of a plane, unit squares with colored edges of p colors are arranged side by side such that adjacent tiles have the same colors. Given a set of Wang tiles B, the nonemptiness problem is to determine whether or not Σ(B) ≠ 0, where Σ(B) is the set of all global patterns on ℤ 2 that can be constructed from the Wang tiles in B. When p > 5, the problem is well known to be undecidable. This work proves that when p = 2, the problem is decidable. is the set of all periodic patterns on ℤ 2 that can be generated by B. If ≠ 0, then has a subset B' of minimal cycle generator such that P(B') ≠ 0 and P(B) = 0 for B ⊂ ≠ B'. This study demonstrates that the set of all minimal cycle generators C(2) contains 38 elements. N(2) is the set of all maximal noncycle generators: if ∈ N(2), then = 0 and ⊃ ≠ implies ≠ O. N(2) has eight elements. That Σ (B) = 0 for any ∈ N(2) is proven, implying that if Σ (B) ≠ 0, then ≠ 0. The problem is decidable for p = 2: Σ(B) ≠ 0 if and only if has a subset of minimal cycle generators. The approach can be applied to corner coloring with a slight modification, and similar results hold.